Find $\dfrac{d}{dx}(-4\cdot5^x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $-20^x\log_5(x)$ (Choice B) B $-20^x\ln(5)$ (Choice C) C $-4\cdot 5^x\ln(5)$ (Choice D) D $-4\cdot 5^x\log_5(x)$
Solution: The expression to differentiate includes an exponential term. Remember that the derivative of the general exponential term $a^x$ (where $a$ is any positive constant) is $\ln(a)\cdot a^x$. Put another way, $\dfrac{d}{dx}(a^x)=\ln(a)\cdot a^x$. $\begin{aligned} &\phantom{=}\dfrac{d}{dx}(-4\cdot5^x) \\\\ &=-4\dfrac{d}{dx}(5^x) \\\\ &=-4\cdot\ln(5)\cdot5^x \\\\ &=-4\cdot5^x\ln(5) \end{aligned}$ In conclusion, $\dfrac{d}{dx}(-4\cdot5^x)=-4\cdot5^x\ln(5)$.